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# If 2^N is a factor of a certain number, the greatest value

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Senior Manager
Joined: 08 Aug 2005
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If 2^N is a factor of a certain number, the greatest value [#permalink]  27 Nov 2005, 01:48
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If 2^N is a factor of a certain number, the greatest value of N is the 2-height of the number. Which of M and K has the greater 2-height?

1). K>M
2). K/M is an even number.
SVP
Joined: 24 Sep 2005
Posts: 1891
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Re: DS [#permalink]  27 Nov 2005, 02:46
getzgetzu wrote:
If 2^N is a factor of a certain number, the greatest value of N is the 2-height of the number. Which of M and K has the greater 2-height?

1). K>M
2). K/M is an even number.

1. K=9>M=8
2^0=1 is a factor of K
2^3=8 is a factor of K
---> 2-height of K is smaller than that of M

K=8, M=2
2-height of K is bigger than that of M
---->insuff

2. since 2^N is a factor of number ---> 2^N must be > 0
There're two cases: both K and M are even, K is even and M is odd
in case : both K and M are even:
since K/M= even number ----> K is a multiple of M --> K= M*2l
2-height of M is 2^y----> K= 2^y * t (t is integer) * 2l = tl* 2^(y+1)
-->2-height of K is y+1, that of M is y ---> 2-height of L is > that of M

in case K is even and M is odd ---> 2-height of M is 0
2-height of K (since K is even) must be >0
----> 2-height of K> that of M

both cases yield the same result --->suff

Hik,i hurry to bathe now ( no more hot water if late) ..it's clumsy , hope you get what i explain ...anyway, i'll re-check my reasoning
Director
Joined: 14 Sep 2005
Posts: 994
Location: South Korea
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2) K/M = 2a (a is a positive integer)

K = M * 2a

K has at least one more 2 than M does.
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Auge um Auge, Zahn um Zahn !

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